Krull dimension In commutative algebra, the Krull dimension…

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Krull dimension
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I.
A field k has Krull dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.
There are several other ways that have been used to define the dimension of a ring. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings.
Explanation[edit]
We say that a chain of prime ideals of the form has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of to be the supremum of the lengths of all chains of prime ideals in .
Given a prime ideal in R, we define the height of , written , to be the supremum of the lengths of all chains of prime ideals contained in , meaning that .[1] In other words, the height of is the Krull dimension of the localization of R at . A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal.
In a Noetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension.[2] A ring is called catenary if any inclusion of prime ideals can be extended to a maximal chain of prime ideals between and , and any two maximal chains between and have the same length. A ring is called universally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.[3]
In a Noetherian ring, a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by n elements (Krull's height theorem and its converse).[4] It implies that the descending chain condition holds for prime ideals in such a way the lengths of the chains descending from a prime ideal are bounded by the number of generators of the prime.[5]
More generally, the height of an ideal I is the infimum of the heights of all prime ideals containing I. In the language of algebraic geometry, this is the codimension of the subvariety of Spec() corresponding to I.[1]
Schemes[edit]
It follows readily from the definition of the spectrum of a ring Spec(R), the space of prime ideals of R equipped with the Zariski topology, that the Krull dimension of R is equal to the dimension of its spectrum as a topological space, meaning the supremum of the lengths of all chains of irreducible closed subsets. This follows immediately from the Galois connection between ideals of R and closed subsets of Spec(R) and the observation that, by the definition of Spec(R), each prime ideal of R corresponds to a generic point of the closed subset associated to by the Galois connection.
Examples[edit]
- The dimension of a polynomial ring over a field k[x1, ..., xn] is the number of variables n. In the language of algebraic geometry, this says that the affine space of dimension n over a field has dimension n, as expected. In general, if R is a Noetherian ring of dimension n, then the dimension of R[x] is n + 1. If the Noetherian hypothesis is dropped, then R[x] can have dimension anywhere between n + 1 and 2n + 1.
- For example, the ideal has height 2 since we can form the maximal ascending chain of prime ideals.
- Given an irreducible polynomial , the ideal is not prime (since , but neither of the factors are), but we can easily compute the height since the smallest prime ideal containing is just .
- The ring of integers Z has dimension 1. More generally, any principal ideal domain that is not a field has dimension 1.
- An integral domain is a field if and only if its Krull dimension is zero. Dedekind domains that are not fields (for example, discrete valuation rings) have dimension one.
- The Krull dimension of the zero ring is typically defined to be either or . The zero ring is the only ring with a negative dimension.
- A ring is Artinian if and only if it is Noetherian and its Krull dimension is ≤0.
- An integral extension of a ring has the same dimension as the ring does.
- Let R be an algebra over a field k that is an integral domain. Then the Krull dimension of R is less than or equal to the …